Many practical processes are influenced by uncertainties, which might have
a large impact. Therefore, these uncertainties should be considered when
optimizing such process. One approach of incorporating uncertain influences
is the usage of chance constrained optimization. This approach requires
that the constraints are only held with a certain probability level,
thereby allowing a compromised decision between reliability and
profitability.
Depending on the underlying process, several approaches to transform the
chance constraints into deterministic constraints exist. Most of these
approaches are based on high-dimensional integrals. In this work,
corresponding methods for the evaluation of such integrals will be
introduced. In doing so, the focus is always on efficient numerical
implementations. An essential part of this thesis is the characterization
of so called analytical approximations, which can be efficiently used for a
large class of applications. For these approaches, methods to evaluate
gradients are described. A further reduction of the computation time can be
achieved through an efficient approximation of the underlying model
equations.
In the case studies, the analytical approximations are compared with
several other approaches. One result is that analytical approximations can
act as general purpose approaches, although other methods lead to slightly
better optimization results. The largest case study deals with a problem
from the area of optimal power flow. Here, it can be shown that the results
obtained by the proposed approach is better than the results obtained
through the usage of the widely employed Gaussian approximation.
Furthermore, by using efficient methods even larger scale case studies can
be solved in reasonable computation time.